# Calc07_4 - 7.4 Lengths of Curves and Surface Area(Photo not...

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7.4 Lengths of Curves and Surface Area Greg Kelly, Hanford High School, Richland, Washington (Photo not taken by Vickie Kelly)

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If we want to approximate the length of a curve, over a short distance we could measure a straight line. ds dx dy By the pythagorean theorem: 2 2 2 ds dx dy = + 2 2 ds dx dy = + 2 2 ds dx dy = + We need to get dx out from under the radical. 2 2 2 2 2 dx dy S dx dx dx = + 2 2 1 dy L dx dx = + 2 1 b a dy L dx dx = + Length of Curve (Cartesian) Lengths of Curves:
2 9 y x = - + 0 3 x Example: 2 9 y x = - + 2 dy x dx = - 2 3 0 1 dy L dx dx = + ( 29 3 2 0 1 2 L x dx = + - 3 2 0 1 4 L x dx = + Now what? This doesn’t fit any formula, and we started with a pretty simple example! ( 29 ln 37 6 3 37 4 2 L + = + 9.74708875861 The TI-89 gets:

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2 9 y x = - + 0 3 x Example: ( 29 ln 37 6 3 37 4 2 L + = + 9.74708875861 2 2 2 9 3 C + = 2 81 9 C + = 2 90 C = 9.49 C The curve should be a little longer than the straight line, so our answer seems reasonable.
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